This is strange because the reduced composite model should have fewer df (fewer restrictions) and should normally fit at least as good as the latent multiple indicator model. I would look for some kind of error or omitted path/effect in the specification.

Also, N = 300 is not so small and the fit of your original model seems decent. I would consider using the original model.

Thank you so much, Christian Geiser

I used to think the same but there is no clue in the literature how could CFI and TLI are not at all correlated. I also tested omitted path effect but it does not change the model fit indices much. I and my co-authors also feel the sample size is not terribly bad for the model, but unfortunately, the reviewer says it is too small for our complex model.

Hi,

I am not 100% sure, but these are my thoughts on that issue.

The CFI and TLI are computed as follows (see http://www.davidakenny.net/cm/fit.htm)

CFI = ((Chi_2_H0 – df_h0) – (Chi_2_H1 – df_h1))/(Chi_2_H0 – df_h0)

TLI = ((Chi_2_H0 /df_h0) – (Chi_2_H0 / df_h0))/(Chi_2_H0 /df_h0)

Thus, CFI and TLI penalty complexity differently.

Your model appears almost saturated (df = 1). I simulated a model similar to yours (see attached). Note, that a guessed parameters blindly to reproduce the discrepancy between CFI and TLI.

In the first (miss-specified) model, I fixed the path between Y4 and X2 to zero. The misfit was similar to that you observed (Chi (df = 1) = 9.257, CFI = .960, TLI = .435). The model tells that omitting Y4 and X2 is problematic and that the association between Y4 and X2 is statistically significant.

In the second model, I additionally constrained the path between Y4 and X1 to zero. However, I simulated data such that Y4 ON X1 is statistically insignificant. Thus, adding that constrained had almost no impact on the Chi-Square value (9.398, df = 2); however, the TLI appears more favorable, while the CFI remained almost unchanged (TLI = .747, CFI = .964).

Plugging the values into the formula of CFI and TLI and computing the ratio of the numerators reveals that the extra degree of freedom has a much stronger influence on the TFI (considering that the nominator is the same in both models).

Ratio of numerators CFI = (((218.637 - 14)-(9.398-2)))/(((218.637 - 14)-(9.257-1)) ~ 1 Ratio of numerators TFI = (((218.637/14)-(9.398/2)))/(((218.637/14)-(9.257/1))) ~ 1.7

This is of course just a description of what happens within the simulated dataset and not a general rule. However, this example illustrates that the discrepancy you observed is probably the result of model with low df that inappropriately omits a statistically significant path between Y4 and X4.

However, this is just guessing because you did not show the full model.

Hope that helps, Manuel

Wow, this is super helpful. Thank you so much, Manuel Heinrich